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16 Hudson Valley Parent ■ September 2014 This is where the early exposure to number bonds is useful: a student who thinks of a number like 12 as something that can be broken apart into 10 + 2 (or 7 + 5 or any other combination) will have no diffi culty seeing a number like 23 as 20 + 3, and 37 as 30 + 7. This allows them to divide the rectangle as: By breaking 37 and 23 apart in this fashion, we have four rectan- gles whose areas are easy to fi nd, because their lengths and widths are numbers with a single non-zero dig- it. Thus the rectangle in the upper left has width 30 and length 20, for area 30 × 20 = 600; the rectangle in the upper right has width 7 and length 20, for area 7 × 20 = 140; the rectangle in the lower right has area 30 × 3 = 90; and the rectangle in the lower left has area 7 × 3 = 21. Adding these four areas together gives us the product: 851. Even larger products can be found using the area model. For example, 356 × 48 can be found by writing 356 as 300 + 50 + 6 and 48 as 40 + 8 (again, retaining the feature that each term has only a Dear Jeff, What's an "area model"? In 4th and 5th grade, students are introduced to the multiplication of multi-digit numbers. One of the more effec- tive means of fi nding these products involves an area model, which connects to the 3rd grade result that the area of a rectangle is the product of its width and length. For example, you can fi nd the product 23 × 37 by fi nding the area of a rectangle with a width of 23 and a length of 37: Of course, there is little gained if you simply multiply 23 × 37. What makes the area model useful is that you are now able to break up one large area (in this case, that of a 23 × 37 rectangle) into several smaller pieces, fi nd the areas of the smaller pieces individually, then add them all together get the area of the whole. single non-zero digit), then fi nding the areas of the six regions. To point out one useful feature of the area model, I've introduced an error in the computation: Now consider the standard algo- rithm (also introduced in the 5th grade) for this product; again, I've introduced an error in the computa- tion: 3 5 6 × 4 8 2 8 4 8 1 4 7 4 1 7 5 8 8 The standard algorithm steps have no justifi cation beyond "Do as you're told," so it's easy for students to lose track of what they're doing and make mistakes, and hard to fi nd and fi x any mistakes that they make. In contrast, the area model is much more transparent, so it's easier for students to keep track of what they're doing, they're less likely to make mistakes, and it's easier for them to fi nd and fi x mistakes. Try fi nding the error in the area model computation, and the error in the standard algorithm computation; then see how easy it is to fi x these errors in the respective algorithms. Jeff Suzuki teaches mathematics at Brooklyn College. Common Core @ Home Only at hvparent.com: Dear Kiersten, My son is entering the 3rd grade this year. How can I fi nd out more about the Com- mon Core curriculum? Answers from Common Core experts Part Seven of a Continuing Series JEFF SUZUKI Mathematics

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